Markov Processes, Hurst Exponents, and Nonlinear
Diffusion Equations
with application to finance
Kevin E. Bassler, Gemunu H. Gunaratne+, & Joseph L. McCauley++
Physics Department
University of Houston
Houston, Tx. 77204
jmccauley@uh.edu
+
Institute of Fundamental Studies
Kandy, Sri Lanka
++
Senior Fellow
COBERA
Department of Economics
J.E.Cairnes Graduate School of Business and Public Policy
NUI Galway, Ireland
Key Words: Hurst exponent, Markov process, scaling, stochastic
calculus, autocorrelations, fractional Brownian motion, Tsallis model,
nonlinear diffusion
Abstract
We show by explicit closed form calculations that a Hurst exponent
H≠1/2 does not necessarily imply long time correlations like those
found in fractional Brownian motion. We construct a large set of
scaling solutions of Fokker-Planck partial differential equations
where H≠1/2. Thus Markov processes, which by construction have
no long time correlations, can have H≠1/2. If a Markov process scales
with Hurst exponent H≠1/2 then it simply means that the process
has nonstationary increments. For the scaling solutions, we show
how to reduce the calculation of the probability density to a single
integration once the diffusion coefficient D(x,t) is specified. As an
example, we generate a class of student-t-like densities from the class
of quadratic diffusion coefficients. Notably, the Tsallis density is one
member of that large class. The Tsallis density is usually thought to
result from a nonlinear diffusion equation, but instead we explicitly
show that it follows from a Markov process generated by a linear
Fokker-Planck equation, and therefore from a corresponding
Langevin equation. Having a Tsallis density with H≠1/2 therefore
does not imply dynamics with correlated signals, e.g., like those of
fractional Brownian motion. A short review of the requirements for
fractional Brownian motion is given for clarity, and we explain why
the usual simple argument that H≠1/2 implies correlations fails for
Markov processes with scaling solutions. Finally, we discuss the
question of scaling of the full Green function g(x,t;x’,t’) of the FokkerPlanck pde.
1.
Introduction
Hurst exponents are widely used to characterize stochastic processes,
and are often associated with the existence of auto-correlations that
describe long term memory in signals [1]. In finance they are used as
a measure of the “efficiency” of markets where a value of the Hurst
exponent H=1/2 is often said to be required by the efficient market
hypothesis (EMH). In this paper we explicitly demonstrate that
H≠1/2 is consistent with Markov processes, which by construction
have no memory. Therefore, we show that the Hurst exponent alone
cannot be used to determine either the existence of long term
memory or the efficiency of markets.
As originally defined by Mandelbrot [1], the Hurst exponent H
describes (among other things) the scaling of the variance of a
stochastic process x(t),
∞
σ 2 = ∫ x2 f(x, t)dx = ct 2H
−∞
€
(1)
where c is a constant. Here, the initial point in the time series xo=0 is
assumed known at t=0. Initially, we limit our analysis to a drift-free
process, so that <x>=0.
A Markov process [2,3] is a stochastic process without memory: the
conditional probability density for x(tn+1) in a time series {x(t1), x(t2),
… x(tn)} depends only on x(tn), and so is independent of all earlier
trajectory history x1, …, xn-1. In financial applications our variable x
should be understood to be the logarithmic return x(t)=lnp(t)/po
where p(t) is the price of a financial instrument at time t. A stochastic
differential equation
dx = D(x, t)dB(t)
€
(2)
describes a Markov process, where B(t) is the Wiener process [4] with
<dB>=0, <dB2>=dB2=dt (dB/dt is white noise). That (2) is Markovian
is proven very abstractly in the literature [4,5], so we provide the
reader with a more straightforward argument in part 3.
Consider next what is required in order to obtain (1), namely, a
probability density f(x,t) that scales with a Hurst exponent H, 0<H<1,
f(x, t) = t −H F(u);u = x/t H . (3)
€
The scaling form (3) guarantees that the variance (1) scales. For a
Markov process one must also be able to calculate the variance from
the sde (2) as
t
∞
0
−∞
σ 2 = ∫ ds ∫ dxf(x, s)D(x, s), (1b)
(where we use Ito calculus [4,5]) so that diffusion coefficient scaling
€
D(x, t) = t 2H−1 D(u),u = x/t H (4)
is a consequence of (3) as well.
€
The conjecture that H≠1/2 implies correlations follows from the
following simple argument [6,7]. Calculate the autocorrelation
function in the following way:
2 Δx(t − Δt)Δx(t + Δt) = (Δx(t − Δt) + Δx(t + Δt))2 − Δx2 (t − Δt) − Δx2 (t + Δt)
(5)
€
If the stochastic process x(t) has stationary increments [2], requiring for
one thing that the mean square fluctuation calculated from any
starting point x(t) scales,
(x(t + Δt) − x(t))2 = cΔt 2H
€
,
(6)
because it depends only on Δt and not on t, then rescaling the
autocorrelation function by the mean square fluctuation,
C(-Δt, Δt)=< Δx(t-Δt)Δx(t+Δt)>/< Δx2(Δt)>, we have
C(−Δt, Δt) = 2 2H−1 − 1 (7)
€
so that H≠1/2 implies autocorrelations. This is the likely origin of the
common expectation that H≠1/2 violates the condition for both a
Markov process and the EMH [6].
However, if (7) would hold for Markov processes then scaling
solutions of the form (3,4) could not exist for those processes (2). But
we will show in part 4 by direct construction that such solutions do
exist, and will show in part 5 that the step from (5) to (7) is doesn’t
hold for a Markov process with H≠1/2, so that when (5) is calculated
directly, then the right hand side vanishes.
This means that an empirical measurement or theoretical prediction
of a Hurst exponent, without showing in addition that the increments
are stationary or else that the dynamics actually has memory, cannot
be interpreted as evidence for autocorrelations (7) in data.
We began this project initially with the aim of understanding where
and how the statistics generated by the drift-free nonlinear diffusion
equation
∂fq
∂t
€
=
1 ∂2
2
2 ∂x
(fq
2−q
)
(8)
fit into the theory of stochastic processes. The reason for that
motivation is that the assumption is made in many papers on Tsallis
model dynamics [8,9,10,10b,10c,11] that the nonlinear diffusion eqn.
(8) is a “nonlinear Fokker-Planck” partial differential equation (pde)
with underlying stochastic differential equation (sde)
dx = D(x, t)dB(t) . (2)
€
The standard assumption is that (2) should describe the same process
as the solution fq(x,t) of (8) if we use the diffusion coefficient
1− q
D q (x, t) = fq
€
(x, t)
(9)
in the sde (2). But the question of consistency arises when one tries to
construct a Langevin description (2) for a truly nonlinear diffusion
pde (we state the inconsistency explicitly in part 3 below). The
question left unanswered in all the papers on the Tsallis model is:
what is the density of f(x,t) of x that is generated by (2) if we use (9),
where fq solves the pde (8)? So far, no proof has been given that the
two separate methods yield the same probability density in the many
papers written while making that assumption. It has also been
assumed [8] that the Hurst exponent H≠1/2 in fq signals fractional
Brownian motion (fBm). We want to analyze these assumptions
carefully. Toward that end, we explain in part 3 why (2) is always
Markovian. The Tsallis model solution fq of (8) is exhibited in part 6.
Finally, by studying the one parameter class of quadratic diffusion
coefficients, D(u)=(1+εu2) we generate the entire class of student-tlike densities. Student-t distributions have been used frequently in
biology and astronomy as well as in finance, so that the diffusion
processes analyzed in part 4 will be of interest in those fields. Since
the processes are Markovian there is no autocorrelation between
different increments Δx for nonoverlapping time intervals, but it is
well known that there is a form of ‘long range dependence’ that
appears in products of absolute values or squares of random
variables [12]. The discussions in [12] have been restricted to near
equilibrium stochastic processes (asymptotically stationary
processes) like the Ornstein-Uhlenbeck model, and so we plan to
discuss the generalization to scaling solutions (3) in a future paper.
Because the term ‘stationary’ is used differently and is not always
clearly defined in some of the recent literature, we next define
‘stationary’ and ‘nonstationary’ processes as used in this paper.
2. Stationary & Nonstationary Processes, and Processes
with Stationary Increments
Here, we define exactly what we mean by a stationary process
[2,3,13] and also define a process with stationary increments [2].
Consider a stochastic process x(t) with probability density f(x,t). By
stationarity, many authors mean that x(t+Δt)-x(t) depends only on Δt
and not on t, but this is the standard definition [2] of a nonstationary
process with ‘stationary increments’. A stationary process is one
where f(x,t)=f(x) is independent of t [2,3,13]. These processes describe
statistical equilibrium and steady states because averages of
dynamical variables are time-independent. In particular, the mean
and variance of a stationary process are constants. An example of an
asymptotically stationary process is the Ornstein-Uhlenbeck process
dv = −βv + σ 1 dB(t) (10)
€
describing the approach of the velocity distribution f(v,t) of a
Brownian particle to statistical equilibrium (the Maxwell-Boltzmann
velocity distribution) [3].
An example of a nonstationary process that generates stationary
increments is the Green function
g(x, t; x′, t ′) =
1
e−( x− x′) /2DΔt
4πDt
(11)
2
for the simplest diffusion equation
€
€
∂f D ∂2 f
=
∂t 2 ∂x2 (12)
where x is unbounded. Note that (12) also has a (nonnormalizable)
stationary solution, f(x)=ax+b, but that that solution is not
approached dynamically (is not reached by (11)) as t goes to infinity.
If x is instead bounded on both sides (particle in a box), then we get
an asymptotically stationary process: f(x,t) approaches a timeindependent density f(x) as t goes to infinity [2,3]. Stationary
processes are discussed by Heyde and Leonenko [12], who label them
“strictly stationary”. They also discuss processes with and without
stationary increments. Some authors describe a process as ‘weakly
stationary’ if only the mean and variance are constants.
If any moment (mean, variance, etc.) of f(x,t) depends on either t or Δt
then the process is by definition nonstationary [2,3,13]. An example is
any scaling solution f(x,t)=t-HF(u), u=x/tH, of the Fokker-Planck
equation
∂f(x, t) 1 ∂2
=
(D(x, t)f(x, t)) (13)
2
∂t
2 ∂x
€
Here, the variance σ 2=ct2H is always strongly time-dependent, a
scaling solution f(x,t) describes a far from equilibrium process.
Although F(u) is stationary in u, F(u) obeys no Fokker-Planck
equation (correspondingly, the variable u obeys no Langevin
equation). The function F(u) is simply the scale invariant part of a
nonstationary scaling solution f(x,t) of the sde (2).
Nonstationary processes x(t) may have either stationary or
nonstationary increments. A nonstationary process with stationary
increments is defined by x(t+T)-x(t)=x(T). An example of a
nonstationary process with scaling plus stationary increments is
Mandelbrot’s model of fractional Brownian motion [1], which we
exhibit in part 5 below. With stationary increments, not only the
variance (1) scales but we obtain the nontrivial condition
x((t + T) − x(t))2 = cT2H . (14)
as well. Nonstationary increments are generated when
x(t+T)-x(t)≠x(T), and we also provide an example of this in part 5.
€
These definitions are precise and differ from the terminology used in
some of the modern literature, but are consistent with the dynamical
idea of stationary as ‘not changing with time’. ‘Stationary state’
means statistical equilibrium or a constantly driven steady state, so
that averages of all dynamical variables are time-independent
[2,3,13]. Fluctuations near thermodynamic equilibrium are
asymptotically stationary, e.g. For stationary processes, e.g. (10) for
larger t, the fluctuation-dissipation theorem [14] holds so that the
friction constant β describing regression toward equilibrium in (10) is
determined by equilibrium parameters (the temperature of the heat
bath).
3. Stochastic Differential Equations generate Green
Functions for Linear Fokker-Planck PDEs
The proof that the sde (2) is Markovian is not presented transparently
in [4,5], so we present a simplified argument here for completeness.
Our end result will be the (linear) Fokker-Planck pde
∂g 1 ∂2 (Dg)
=
∂t 2 ∂x2
€
where D(x,t) depends on (x,t) alone, and which describes a Markov
process [2,3,4] via it’s Green function, g(x,t;xo,to), where
g(x,t;xo,t)=δ(x-xo). The Green function is the transition rate density
(conditional probability density) for the Markov process. Following
Schulten [15], we use Ito calculus to show that a stochastic differential
equation, or sde
dx =
€
(15)
D(x, t)dB (2)
necessarily generates a conditional probability density g(x,t;xo,to)
which, by construction, is the Green function for the Fokker-Planck
pde (15),
g(x, xo ; t, t o ) = δ(Δx − D • ΔB) =
€
1 ∞ ikΔx −i
e
∫e
2π =∞
D•ΔB
dk
. (16)
where the dot denotes the Ito product, the stochastic integral of
√D(x,t) with respect to B [4]. In all that follows we use Ito calculus
because Ito sdes are one to one with (linear) Fokker-Planck pdes, as
we will show below. In (16), the average is over all Wiener processes
B(t), so that the Green function can be written as a functional integral
over locally Gaussian densities with ‘local volatility’ D(x,t) [16]. By
f(x,t) in this paper, we mean the Green function f(x,t)=g(x,0;t,0) for
the special case where xo=0 at to=0. Next, we show why the FokkerPlanck pde (15) is required by the sde (2) (it is well known that the
two equations transform one to one under coordinate
transformations whenever Ito calculus is used), and why the FokkerPlanck pde must be linear. The connection between Wiener integrals
for stochastic processes and linear diffusive partial differential
equations was first discussed by Kac (see ch. 4 in [17]), but see also
Durrett [5] for interesting examples of ‘solving a (parabolic) pde by
running a Brownian motion (sde)’
Beginning with the sde (2) but with drift included,
dx = R(x, t)dt + D(x, t)dB , (17)
€
consider the time evolution of any dynamical variable A(x) that does
not depend explicitly on t (e.g., A(x)=x2). The sde for A is given by
Ito’s lemma [4,5,15],
 ∂A 1 ∂2 A 
∂A
dA =  R
+
dt +
D(x, t)dB
2 
∂x
 ∂x 2 ∂x 
. (18)
Forming the conditional average
€
€
 ∂A
1 ∂2 A 
dA =  R
+
dt
∂x
2 ∂x2 

(19)
by using the Green function g(x,t;xo,to) generated by (17) and
integrating by parts while ignoring the boundary terms1, we obtain
∂g ∂(Rg) 1 ∂2 (Dg)
−
∫ dxA(x) +
= 0
∂x
2 ∂x2  , (20)
 ∂t
so that for an arbitrary dynamical variable A(x) we get the Fokker-
€
Planck pde
∂g
∂(Rg) 1 ∂2 (Dg)
=−
+
. (21)
∂t
∂x
2 ∂x2
1
If the density g has fat tails, then higher moments will diverge. There, one must
be more careful with the boundary terms.
€
That is, an sde (2) with drift R(x,t) and diffusion D(x,t) generates a
(generally nonGaussian) random time series {x(t)} whose histograms
at different times t are distributed according to the Green function g
of the Markovian linear pde (21), where g(x,t;xo,t)=δ(x-xo). The point
here is that Langevin equations generate exactly the same statistics as
the corresponding linear Fokker-Planck pde.
In what follows we will assume that R may depend on t but not on x,
so that we can replace x by x- ∫Rdt in (21) to get the drift free pde (15).
An x-dependent drift R(x,t) is inconsistent with the scaling form
(3).
At this point we can make a prediction: either the nonlinear pde (8)
has no underlying Langevin equation (2), because a nonlinear pde
has no Green function and the Green function (transition probability
density) is the heart and soul of a Markov process. Or, the pde (8) is
really a linear Fokker-Planck pde somehow disguised as a nonlinear
diffusion equation.
4. Markov Processes with Scaling Solutions
Until the last section of this paper, we restrict to the case where
g(x,0,;t,0)=f(x,t) because we will show explicitly below that only these
solutions exhibit exact scaling properties (1), (3), (4), and also because
the density f(x,t) is what one observes directly in histograms of
finance market returns data. The full Green function g(x,x’;t,t’) is
needed in principle for exact option pricing (see [16,21] for another
requirement) but cannot be calculated in closed analytic form when
D(x,t) depends on both x and t and scaling (3, 4) holds. If f and
therefore D scale according to (3) and (4), then the variance scales
exactly as (1) with Hurst exponent H. The empirical evidence for the
data collapse predicted by (3) will be presented in a separate paper
on financial markets [18]. The question of scaling of the full Green
function is discussed at the end of this paper.
Inserting the scaling forms (3) and (4) into the pde (15), we obtain2
2
We emphasize that the drift has been subtracted out of the pde (21) to yield (15).
This requires that R(x,t) is independent of x. A x-independent drift R(t) is
absolutely necessary for the scaling forms (3,4).
2H(uF(u))′ + (D(u)F(u))′′ = 0 (22)
whose solution is given by
€
€
F(u) =
C −2H ∫ udu/D(u)
.
e
D(u)
(23)
Note that (23) describes the scale invariant part of nonstationary
solutions f(x,t) (3). This generalizes our earlier results [19,20,20b] to
include H≠1/2.
Next, we study the class of quadratic diffusion coefficients
D(u) = d ′(ε)(1 + εu2 ), (24)
which yields the two parameter (ε,H) class of student-t- like densities
€
F(u) = C′(1 + εu2 )−1− H / εd′(ε) (25)
€
with tail exponent µ=2+2H/εd ’ ( ε ) , and where H and ε are
independent parameters to be determined empirically. This shows
where student-t-like distributions come from. With d’(ε)=1 we obtain
the generalization of our earlier prediction [20] to arbitrary H, 0<H<1,
with µ=2+2H/ε. Here, we generate all fat tail exponents 2<µ<∞, and
obtain a finite variance (1) scaling as σ 2=ct2H whenever µ>3 (for
2≤µ≤3 the variance is infinite). For large u this model fits the fat tails
in financial data for all times t [18]. For small to moderate returns
finance market histograms are approximately exponentially
distributed [16,21], and a complete empirical analysis of market data
will be presented in a forthcoming separate paper [18].
5. What does H mean?
The assumption that “H≠1/2 implies long time correlations” is often
found in the literature (see, however, Mandelbrot [22]). An example
where H≠1/2 but there are no autocorrelations is the Levy density
[23]. In that case, H describes the scaling of the peak of the density
and also the tail exponent. For the Levy densities the variance is
infinite.
Consider the Markov process (2) in its integrated form
t +Δt
Δx(t, t + Δt) = ∫
D(x(s), s)dB(s)
(26)
t
€
Even when scaling (3,4) holds with H≠1/2 , then on quite general
grounds there can be no autocorrelation in the increments Δx(t,t-Δt),
Δx(t,t+Δt) over two nonoverrlapping time intervals
[t, t + Δt]∩[t − Δt, t] = ∅ .
(27)
This is easy to see: by definition of the Ito integral:
€
(28)
€
t +Δt
t
t
t−Δt
Δx(t, t − Δt)Δx(t, t + Δt) = ∫ ds ∫ dw D(x(w), w)D(x(s), s) dB(w)dB(s) = 0
because <dB(w)dB(s)>=0 for nonoverlapping time intervals dw and
ds [4,5,15]. The function D(x,t) is called ‘nonanticipating’ [4]. This just
means that, by Ito’s definition of the stochastic integral (26), the
function D(x,t) of random variable x and the random increment dB(t)
from t to t+dt are statistically dependent because x(t) was determined
in the sde (2) by the Wiener increment dB(t-dt) before dB(t) occurs.
That is, D(x(t),t) cannot ‘anticipate’ the next random increment dB(t)
in (26).
The passage from (5) to (7) requires a usually unstated assumption of
stationary increments. If the nonstationary stochastic process x(t) has
nonstationary increments, requiring that the mean square fluctuation
about x(t) depends both on Δt and t, then the passage from (5) to (7) is
not possible. The argument that H≠1/2 implies long time
correlations fails for Markov processes precisely because the
stochastic integral (26) with the scaling forms (3,4) describes a
nonstationary process with nonstationary increments whenever H≠1/2.
Only for H=1/2 do we retrieve a nonstationary process with
stationary increments. When H≠1/2 then (26) combined with scaling
(4) yields (with x(0)=0)
t +T
x(t+ T) − x(t) = ∫
0
t +T
= ∫
t
T
€
0
t +T
D(x(s),s) dB(s) = ∫ s
= ∫ s+t
0
t
D(x(s),s) dB(s) − ∫ D(x(s), s) dB(s)
H−1/2
D(u) dB(s)
t
H−1/2
H
D(x/ s + t ) dB(s + t) ≠ x(T)
(26b)
whereas we retrieve stationary increments x(t+T)-x(t)=x(T) for H=1/2
with probability one, e.g., with H=1/2 we find that
<(x(t+T)-x(t))2>=<x2(T)>=cT. Furthermore, direct calculation of the
autocorrelation formulated as (5) shows that the right hand side of (5)
vanishes independently of the value of H, in agreement with (28)
above.
We’ve seen above that a Hurst exponent H≠1/2 is consistent with a
Markov process. One only needs the scaling forms (3,4), and the
Fokker-Planck pde (15) is then satisfied by f(x,t)=t-HF(u) with u=x/tH
where F(u) is given by (23). This Hurst exponent does not imply long
time correlations, so what does H≠1/2 mean? The appearance of
H≠1/2 in a Markov process signals underlying dynamics with
nonstationary increments, and this knowledge should be useful for
data analysis.
From a purely theoretical standpoint, a Hurst exponent H≠1/2 for a
scale free Markov process can be eliminated by a change of time
variable (a corollary is that any Markov process with H=1/2 can be
converted superficially into one with H≠1/2 by a change of time
scale). Note that for any diffusion coefficient of the form
D(x,t)=h(t)d(x,t), the prefactor h(t) can always be absorbed into a
redefinition of the time scale in the drift-free Fokker-Planck pde (15),
dτ=h(t)dt. Likewise, with the choice of time variable τ=tH, the pde
(15) with the scaling forms (2) and (1) always yields σ2=cτ. So a drift
free Markov process with nonstationary increments can be
transformed formally into one with stationary increments by the
appropriate change of time scale, and vice-versa.
There can be no correlations for nonoverlapping time intervals because (26)
is Markovian, whether H=1/2 or H≠1/2 plays no role. This is why
Markov dynamics reflect the EMH: a Markovian market is impossible
to beat. Real markets are very hard to beat systematically over the
long haul, so that a Markov model provides us with a very good
zeroth order approximation to real financial markets. Another way to
say it is, with drift subtracted out, a market is pure (nonGaussian)
noise, in agreement with Black’s idea of the importance of ‘noise
traders’ [24]. When H≠1/2 combined with stationary increments in
x(t) then there is either persistence or antipersistence of
autocorrelations for nonoverlapping time intervals, as in fractional
Brownian motion [1]. Fractional Brownian motion (fBm) is inherently
nonMarkovian. In principle, a market with H≠1/2 plus stationary
increments has correlations that may be exploited for profit, so that
such a market is not “efficient”.
One can construct models of fractional Brownian motion as follows.
With k(t,s)=tH-1/2K(u), u=t/s, a stochastic process of the form
t
xH (t) = ∫ k(t, s)dB(s)
to
€
(29)
generates long time autocorrelations for nonoverlapping time
intervals but doesn’t scale. Scaling is obtained iff. to=0 or -∞. For the
former case the increments of (29) are not stationary, but one may
obtain stationary increments for to = -∞, depending on the form of the
function k(t,s). In that case, we have the scaling law σ2=<x2>=ct2H. If
the kernel k(t,s) is such that xH(t) has stationary increments [1],
T
xH (t + T) − xH (t) = ∫ k(T, s)dB(s) = xH (T)
−∞
€
, (30)
then a simple prediction (a generalization of (7)) for the
autocorrlelations of fBm over nonoverlapping time intervals follows:
with the autocorrelation function defined more generally by
2
C(S1 , S2 ) = (x H (t + Δt1) − x H (t))(x H (−t) − x H (−t − Δt 2 )) / σ1 σ2
2
(31)
where S1=Δt1/t, S2=Δt2/t, we obtain [1]
€
C(S1 , S2 ) = [(1 + S1 + S2 )2H + 1 − (1 + S1)2H − (1 + S2 )2H ] / 2(S1S2 )H
(32)
€
This prediction can easily be generalized to allow widely separated
time intervals [t1-Δt1,t2+Δt2) where t1<t2. Mandelbrot [1] has provided
us with an illuminating example of fBm with stationary increments
(25),
0
t
−∞
0
x H (t) = ∫ [(t − s)H−1 / 2 − (−s)H−1 / 2 ]dB(s) + ∫ [(t − s)H−1 / 2 dB(s)
(29b)
€
Clearly, such long time correlations are nonMarkovian and violate
the EMH. Note that the correlations (32) vanish if H=1/2, whereas in
a Markov process the correlations vanish for all values of H.
Contrary to statements [6,7] in the literature, (29b) is not a Gaussian
process. With D(u)=1 (23) becomes F(u)=exp(-u2) with u=x/tH, but
this solution of (15) describes a Markov process with no
autocorrelations at all. One cant rightly state that (29b) arises from
Gaussian increments dB(t), but then so does every other stochastic
process described by an Ito integral, and those processes are typically
far from Gaussian distributed, as is (29b). We do not yet know the
functional form f(xH,t) of the density of (29b), other than that it must
scale like (3) and cannot be Markovian.
A Markov process provides a sufficient but not necessary condition
for the EMH. Since finance market data can be described as
approximately Markovian [16,19], to zeroth order, then searching for
fBm or other memory in market data would be a search for a way to
make small profit margins by placing big bets.
6. The Tsallis Density
It is easy to check by direct calculation that a normalized solution of
(8) is given self-consistently by
fq (x, t) = (c(2 − q )(3 − q))−H t −H (1 + (q − 1)x2 /C 2 (q )t 2H )1/(1−q) (33)
with H=1/(3-q), where
€
C(q ) = c(q−1)/2(3−q) ((2 − q)(3 − q ))H
(34)
and
€
∞
c1/2 = ∫ du(1+ (q − 1)u 2 )1/(1−q) (35)
−∞
€
is the normalization constant [11]. Normalization is not
overdetermined because the pde (8) satisfies probability
conservation. The fat tail exponent, f(x,t) ≈ x-µ for x>>1, is µ=2/(q-1).
This model has the constraint that the tail exponent µ is fixed by the
Hurst exponent H, or vice-versa. E.g., if H=1/2, then there are no fat
tails, the density is Gaussian.
Inserting (33) into (9) yields the diffusion coefficient
D q (x, t) = (c(2 − q )(3 − q))2H−1 t 2H−1 (1 + (q − 1)x2 /C 2 (q )t 2H ) (36)
which we conveniently rewrite as
€
€
D q (x, t) = d(q)t 2H−1(1 + ((q − 1) / C2 (q))u 2 ) = t 2H−1 D q (u) (37)
To compare (33) with (25), we need only write ε=(q-1)/C2(q) and
d’(ε)=d(q). Our Fokker-Planck-generated density f(x,t) given by (25)
reduces exactly to (33) when H=1/(3-q). This means that fq actually
satisfies the linear Fokker-Planck pde
2
∂f 1 ∂ (D q f)
=
(8b)
∂t 2 ∂x2
and so (8), for the Tsallis solution (33), is really a linear pde disguised
as a nonlinear one.
€
A nonlinear disguise is possible for our entire two-parameter
student-t-like class solutions (25), because for quadratic diffusion
(24), D(u)=d’(ε)(1+εu2), the solution of the Fokker-Planck pde (8) is a
power of the diffusion coefficient, F(u)=CD(u)-1-H/εd’(ε).
All of these solutions trivially satisfy a modified form of the
nonlinear pde (8), but rewriting (8b) as a nonlinear pde in the case of
quadratic diffusion superficially masks the Markovian nature of
Tsallis dynamics.
The claim made is in Borland [8] and elsewhere that Tsallis model (8)
generates fractional Brownian motion, but this is not correct. The
Tsallis density (33) is Markovian and so cannot describe long-time
correlated signals like fBm. There, H=1/(3-q)≠1/2 merely signals that
the increments x(t) are nonstationary.
In a Langevin/Fokker-Planck approach with x-dependent drift,
Kaniadakis and Lapenta [9] did not reproduce the time dependence
of the Tsallis density (33) with H=1/(3-q). In their formulation using
an x-dependent drift term in the Fokker-Planck pde, they find a timedependent solution that does not scale with a Hurst exponent H.
That is, nonscaling solutions are certainly possible. And as we have
pointed out, scaling of f(x,t) is impossible when the drift depends on
x.
But what about truly nonlinear diffusion? The linear pde (8b) solves a
unique initial value problem, and unique boundary value problems
as well. But we do not know if the nonlinear pde
∂f 1 ∂2 m
=
(f )
∂t 2 ∂x2
€
(8c)
with m≠1 has a unique solution for a specified initial condition f(x,0).
There may be solutions other than the trivial self-consistent solution
(33), and there we cannot rule out the possibility of long time
memory in (8c).
For a discussion of the general properties of nonMarkovian linear
pdes with memory, see [25]. See also Hillerbrand and Friedrich [25b]
for nonMarkov densities of the form f(x,t)=t-3/2F(x/t1/2) based on
memory in the diffusion coefficient.
7. Scaling and the Green function
Finally, a few words about the full Green function of the FokkerPlanck pde (15). So far, we’ve restricted to a special case where
f(x,t)=g(x,t;0,0). In this case, as we’ve shown by direct construction,
the scaling (1,3,4) is exact. For the general Green function g(x,t;x’,t’)
with x’≠0 scaling is not exact and may not exist at all.
If we assume that g(x,t;x’,t’)=g(x,x’;Δt), and if we in addition make
the (unproven) scaling Ansatz
g(x, x′; Δt) = Δt −H G(u, u o ) (39)
€
where u=x/tH, uo=xo/tH, then we would have a mean square
fluctuation
t +Δt
∞
t
−∞
(x − xo )2 = ∫ dsΔs 2H−1 ∫ du(u − u o )2 G(u, u o )
€
(40)
with Δs=s-t. This doesn’t yield a simple expression for nonstationary
increments unless G(u,uo)=G(u-uo), because uo=xo/Δs. We can offer
no theoretical prediction for the Green function when x’≠0.
In a future paper we will analyze option pricing and the construction
of option prices as Martingales, both from the standpoint of
stochastic differential equations [26] and generalized Black-Scholes
equations [16]. A key observation in that case is that, with fat tails,
the option price diverges in the continuum market theory [20b,27].
This result differs markedly from the finite option prices predicted in
[11,28].
8. Summary and Conclusions
Hurst exponents H≠1/2 are perfectly consistent with Markov
processes and the EMH. A Hurst exponent, taken alone, tells us
nothing about autocorrelations. Scaling solutions with arbitrary
Hurst exponents H can be reduced for Markov processes to a single
integration. A truly nonlinear diffusion equation has no underlying
Langevin description. Any nonlinear diffusion equation with a
Langevin description is a linear Fokker-Planck equation in disguised
form. The Tsallis model is Markovian, does not describe fractional
Brownian motion. A Hurst exponent H≠1/2 in a Markov process x(t)
describes nonstationary increments, not autocorrelations in x(t).
Acknowledgement
KEB is supported by the NSF through grants #DMR-0406323 and
#DMR-0427938, by SI International and the AFRL, and by TcSUH.
GHG is supported by the NSF through grant #PHY-0201001 and by
TcSUH. JMC thanks C. Küffner for reading the manuscript and
suggesting changes that made it less redundant and more readable,
and also to Harry Thomas for a careful and critical reading. We're
grateful to a referee for pointing out references 10b and 10c, and to R.
Friedrich for sending us preprint 25b.
References
1.B. Mandelbrot & J. W. van Ness,, 1968. SIAM Rev. 10, 2, 422.
2. R.L. Stratonovich. Topics in the Theory of Random Noise, Gordon &
Breach: N.Y., tr. R. A. Silverman, 1963.
3. N. Wax. Selected Papers on Noise and Stochastic Processes. Dover:
N.Y., 1954.
4. L. Arnold, Stochastic Differential Equations. Krieger, Malabar, Fla.,
1992.
5. R. Durrett, Brownian Motion and Martingales in Analysis,
Wadsworth, Belmont, 1984.
6. B. Mandelbrot, Fractals and Scaling in Finance, Springer, N.Y., 1997.
7. J. Feder. Fractals. Plenum: N.Y., 1988.
8. L. Borland, Phys. Rev. E57, 6634, 1998.
9. G. Kaniadakis & G. Lapenta, Phys Rev E 62,3246, 2000.
10. G. Kaniadakis & P. Quarati, Physica A237, 229, 1997.
10b. G. Kaniadakis, Physica A 296, 405 (2001).
10c. T. D. Frank, Physica A 331, 391 (2004).
11. L. Borland, Quantitative Finance 2, 415, 2002.
12. C.C. Heyde & N.N. Leonenko. Adv. Appl. Prob. 37, 342, 2005.
13. A.M. Yaglom & I.M. Yaglom, An introduction to the Theory of
Stationary Random Functions. Transl. and ed. by R. A. Silverman.
Prentice-Hall: Englewood Cliffs, N.J., 1962.
14. R. Kubo, M. Toda, & N. Hashitsume. Statistical Physics II:
Nonequilibrium Statistical Mechanics. Springer-Verlag: Berlin, 1978.
15.K.Schulten,http://www.ks.uiuc.edu/Services/Class/PHYS498/Lect
ureNotes.html, 1999.
16. J.L. McCauley, Dynamics of Markets: Econophysics and Finance,
Cambridge, Cambridge, 2003.
17. M. Kac,, Probability and related Topics in Physical Sciences, WileyInterscience, N.Y., 1959.
18. A. L. Alejandro-Quinones, K.E. Bassler, J.L. McCauley, and G.H.
Gunaratne, in preparation, 2005.
19. G.H. Gunaratne & J.L. McCauley. Proc. of SPIE conf. on Noise &
Fluctuations 2005, 5848,131, 2005.
20. A. L. Alejandro-Quinones, K.E. Bassler, M. Field, J.L. McCauley,
M. Nicol, I. Timofeyev, A. Török, and G.H. Gunaratne, Physica A,
2005 (in press).
20b. A. L. Alejandro-Quinones, K.E. Bassler, J.L. McCauley, and G.H.
Gunaratne Proc. of SPIE, Vol. 5848, 27, 2005.
21. J.L. McCauley & G.H. Gunaratne, Physica A329, 178, 2003.
22. B. Mandelbrot and J.R. Wallis, Water Resources Research 4, 909,
1968.
23. R. Mantegna, R. & H.E. Stanley. An Intro. to Econophysics.
Cambridge Univ. Pr., Cambridge, 2000.
24. F. Black. J. of Finance 3, 529, 1986.
25. P. Hänggi & H. Thomas, Z. Physik B 26, 85-92, 1977.
25b. R. Hillerbrand & R. Friedrich, Model for the Statistics of Lagrangian
Velocity Increments in Fully Developed Turbulence, preprint (2005).
26. J.M. Steele, Stochastic Calculus and Financial Applications. SpringerVerlag, N.Y., 2000.
27. J. L. McCauley, G.H. Gunaratne, & K.E. Bassler, in Dynmamics of
Complex Interconnected Systems, Networks and Bioprocesses, ed. A.T.
Skjeltorp & A. Belyushkin, Springer, NY, 2005.
28. L. Borland, Phy. Rev. Lett. 89, 9, 2002.